2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/172568In a system made up of independent random walks, fluctuations of order $n^{1/4}$ from the hydrodynamic limit come from particle current across characteristics. We show that a two-parameter space-time particle current process converges to a two-parameter Gaussian process. These Gaussian processes also appear as the limit for the one-dimensional random average process. The final section of this paper looks at large deviations of the current process.to appear in AleaProbability60K35, 60F10text